کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
6422196 1340618 2011 13 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
One-point pseudospectral collocation for the one-dimensional Bratu equation
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات ریاضیات کاربردی
پیش نمایش صفحه اول مقاله
One-point pseudospectral collocation for the one-dimensional Bratu equation
چکیده انگلیسی

The one-dimensional planar Bratu problem is uxx + λ exp(u) = 0 subject to u(±1) = 0. Because there is an analytical solution, this problem has been widely used to test numerical and perturbative schemes. We show that over the entire lower branch, and most of the upper branch, the solution is well approximated by a parabola, u(x) ≈ u0 (1 − x2) where u0 is determined by collocation at a single point x = ξ. The collocation equation can be solved explicitly in terms of the Lambert W-function as u(0) ≈ −W(−λ(1 − ξ2)/2)/(1 − ξ2) where both real-valued branches of the W-function yield good approximations to the two branches of the Bratu function. We carefully analyze the consequences of the choice of ξ. We also analyze the rate of convergence of a series of even Chebyshev polynomials which extends the one-point approximation to arbitrary accuracy. The Bratu function is so smooth that it is actually poor for comparing methods because even a bad, inefficient algorithm is successful. It is, however, a solution so smooth that a numerical scheme (the collocation or pseudospectral method) yields an explicit, analytical approximation. We also fill some gaps in theory of the Bratu equation. We prove that the general solution can be written in terms of a single, parameter-free β(x) without knowledge of the explicit solution. The analytical solution can only be evaluated by solving a transcendental eigenrelation whose solution is not known explicitly. We give three overlapping perturbative approximations to the eigenrelation, allowing the analytical solution to be easily evaluated throughout the entire parameter space.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Applied Mathematics and Computation - Volume 217, Issue 12, 15 February 2011, Pages 5553-5565
نویسندگان
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